Remarks on Type III Unprojection

ABSTRACT

Type III unprojection plays a very important role in the birational geometry of Fano threefolds (cf. Corti et al., 2000; Reid, 2000). According to Reid (2000, p. 43), it was first introduced by A. Corti on his calculations of Fano threefolds of genus 6 and 7.

It seems that at present a general definition of Type III unprojection is still missing. After proving some general facts about residual ideals in Section 2, we propose a definition for the generic Type III unprojection (Definition 3.3), and prove in Theorem 3.5 that it gives a Gorenstein ring.

Communicated by W. Bruns

Keywords:

• Birational geometry
• Fano threefolds
• Gorenstein rings
• Sarkisov program
• Unprojection

Mathematics Subject Classification:

• Primary 13H10
• Secondary 14E30, 14M05

ACKNOWLEDGMENT

I wish to thank Miles Reid, Frank–Olaf Schreyer and Francesco Zucconi for many useful conversations. This work was financially supported by the Deutschen Forschungsgemeinschaft Schr 307/4-2.